Conformal cage-based deformation with polynomial curves

One or more computer storage media having a plurality of executable instructions embodied thereon, which, when executed by one or more processors, cause the one or more processors to perform a method comprising:

• causing display of a two-dimensional (2-D) image and a 2-D cage model in a rest configuration, the 2-D image comprising a plurality of points, each point of the plurality of points corresponding to a location of a corresponding pixel of the 2-D image, and the 2-D cage model defining an outer boundary of at least a portion of the plurality of points of the 2-D image;
• generating curve control points for the 2-D cage model in the rest configuration, each curve control point controlling curvature of a corresponding segment of the 2-D cage model, each corresponding segment of the 2-D cage model corresponding to a polynomial curve defining the outer boundary of the 2-D cage model, the curve control points including:
• an endpoint curve control point at each vertex of vertices of the outer boundary of the 2-D cage model, wherein each vertex of the vertices defines an endpoint of each segment of the outer boundary of the 2-D cage model, and the 2-D cage model only comprising the vertices on the outer boundary of the 2-D cage model;
• and
• one or more intermediate curve control points for at least one segment of the 2-D cage model between corresponding vertices of the at least one segment;
• determining weights for each point in the portion of the plurality of points of the 2-D image within the outer boundary of the 2-D cage model in the rest configuration based on conformal transformation of each point, wherein the weights are based only on a relationship between each point to initial locations of all of the vertices of the 2-D cage model and a total number of the curve control points on all segments of the outer boundary of the 2-D cage model;
• deforming, responsive to receiving a request, the 2-D cage model from the rest configuration to an updated configuration, the updated configuration being different from the rest configuration by changing curvature of a corresponding polynomial curve of the outer boundary of the 2-D cage model in the rest configuration to a corresponding deformed polynomial curve in the updated configuration;
• generating, based on the weights for each point and corresponding locations of the curve control points of the 2-D cage model in the updated configuration, an updated 2-D image by determining a subsequent location for each point in the portion of the plurality of points of the 2-D image within the outer boundary of the 2-D cage model in the rest configuration to update the location of each corresponding pixel with respect to the updated configuration, wherein the subsequent location is computed based only on the weights for each point and the corresponding locations of the curve control points of the 2-D cage model in the updated configuration;
• and
• causing display of the updated 2-D image.

A computing system comprising:

• a processor;
• and
• a non-transitory computer-readable medium having stored thereon instructions that when executed by the processor, cause the processor to perform operations including:
• causing display of an image and a two-dimensional (2-D) cage model in a first configuration, the image comprising a plurality of points, each point of the plurality of points corresponding to a location of a corresponding pixel of the image, the 2-D cage model defining an outer boundary of at least a portion of the plurality of points of the image, the 2-D cage model comprising vertices that define corresponding endpoints of the outer boundary of the 2-D cage model, and the 2-D cage model only comprising the vertices on the outer boundary;
• generating curve control points between the vertices on at least a portion of the outer boundary of the first configuration of the 2-D cage model, the curve control points configured to adjust corresponding curvature of polynomial curves defining the outer boundary of the 2-D cage model;
• preprocessing weights for each point in the portion of the plurality of points of the image based only on a relationship of each point to initial locations of all of the vertices of the 2-D cage model in the first configuration and a total number of the curve control points for all of the polynomial curves defining the outer boundary of the 2-D cage model;
• deforming the 2-D cage model from the first configuration to a second configuration, wherein at least one curve control point is at a different location in the second configuration thereby changing curvature of a corresponding polynomial curve defining a corresponding portion of the outer boundary of the 2-D cage model in the first configuration to a corresponding deformed polynomial curve in the second configuration;
• generating, based on the first configuration and the second configuration of the 2-D cage model, an updated image by performing a conformal transformation at each point in the portion of the plurality of points of the image within the outer boundary of the 2-D cage model in the first configuration to update the location of each corresponding pixel with respect to the second configuration, the conformal transformation of each point computed based only on the weights for each point and the corresponding deformed polynomial curve in the second configuration, wherein the conformal transformation of each point preserves local angles and proportions around each point from the image to the updated image;
• and
• causing display of the updated image.